Question

If $\text{ω}$ is imaginary cube root of unity then the value of $\frac{\left|3 \omega - 2 \omega ^{2}\right| \left|1 - \omega \right|}{\left|3 \omega ^{2} - 2\right| \left|\omega ^{2} - 1\right|}$ is

Answer 1

We have,
$\frac{\left|3 \omega - 2 \omega ^{2}\right| \left|1 - \omega \right|}{\left|3 \omega ^{2} - 2\right| \left|\omega ^{2} - 1\right|}$
Multiply and divide by $\text{ω}$ , then
$=\frac{\left|3 \omega - 2 \omega ^{2}\right| \left|\omega - \omega ^{2}\right|}{\left|3 \omega ^{3} - 2 \omega \right| \left|\omega ^{2} - 1\right|}$
$=\frac{\omega \left|3 - 2 \omega \right| \left|\omega - \omega ^{2}\right|}{\left|3 \cdot 1 - 2 \omega \right| \left|\omega ^{2} - 1\right|}$
$=\frac{\omega \left|\omega - \omega ^{2}\right|}{\left|\omega ^{2} - 1\right|}$
$=\frac{\left|\omega ^{2} - \omega ^{3}\right|}{\left|\omega ^{2} - 1\right|}$
$=\frac{\left|\omega ^{2} - 1\right|}{\left|\omega ^{2} - 1\right|}=1$